I find it as natural as The interior of the house is dirty. Nobody would really say either one that way, but if they did, the two would be equally acceptable.
If one is just talking about needing to clean their living space, one would just say The house is dirty. Context would make it clear that you're talking about needing to vacuum and dust rather than power wash the siding.
On the other hand, if one needed to make that distinction to someone who didn't understand, one would probably say The inside of the house is dirty. Once you start using the words area inside or interior you're getting into awkward speech.